[alanctgardner2]

That's a pretty silly statement. Surely you can prove that 1+1!=3, which is the negative of 1+1=3. You can also prove that pi is not a rational number, 1/2 is not an integer, etc. There's a post on the front page about attempting to prove P!=NP.

[evincarofautumn]

The typical formulations of all of those proofs are by contradiction, though—so it may be worth distinguishing “prove” from “demonstrate by proof”.

[alanctgardner2]

Is there a distinction between proving and demonstrating by proof? I did a quick search (I'm hardly a mathematician), to quote Wikipedia:

> a proof must demonstrate that a statement is always true

It would seem that, if you can demonstrate a statement is true for a given logic, that is proof. Contradiction and exhaustion (which is a bit more controversial, I think), while less tidy than one might expect, are valid techniques for showing that a statement is always true, and therefore are valid ways to construct a proof.

Note: I encourage any and all maths professors to come and tell me off if I'm way off base.